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AbstractCoevolution between hosts and their parasites is expected to follow a range of possible dynamics, the two extreme cases being called trench warfare (or Red Queen) and arms races. Long-term stable polymorphism at the host and parasite coevolving loci is characteristic of trench warfare, and is expected to promote molecular signatures of balancing selection, while the recurrent allele fixation in arms races should generate selective sweeps. We compare these two scenarios using a finite size haploid gene-for-gene model that includes both mutation and genetic drift. We first show that trench warfare do not necessarily display larger numbers of coevolutionary cycles per unit of time than arms races. We subsequently perform coalescent simulations under these dynamics to generate sequences at both host and parasite loci. Genomic footprints of recurrent selective sweeps are often found, whereas trench warfare yield signatures of balancing selection only in parasite sequences, and only in a limited parameter space. Our results suggest that deterministic models of coevolution with infinite population sizes do not predict reliably the observed genomic signatures, and it may be best to study parasite rather than host populations to find genomic signatures of coevolution such as selective sweeps or balancing selection.